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HILBERT SPACE OF ANALYTIC FUNCTIONS ASSOCIATED WITH THE MODIFIED BESSEL FUNCTION AND RELATED ORTHOGONAL POLYNOMIALS

NOBUHIRO ASAI

Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan

Current address: Department of Mathematics, Meijo University, Nagoya 468-8502 Japan.

https://doi.org/10.1142/S0219025705002098Cited by:6 (Source: Crossref)

Abstract

Let μ be a probability measure on I ⊂ ℝ with finite moment of all orders and γ_{α, c_{1, 2}} be a probability measure on ℂ which is expressed by the modified Bessel function. In this paper, we shall first consider operators B^-, B⁺, B° and express them in terms of bosonic creation and annihilation operators. Secondly, we shall construct the Hilbert space of analytic L² functions with respect to γ_{α, c_{1, 2}}, . It will be seen that the S_μ-transform under certain conditions is a unitary operator from L² (I, μ) onto . Moreover, we shall give explicit examples including Laguerre, Meixner and Meixner–Pollaczek polynomials and explain how the Hilbert space and B^-, B⁺, B° are related from the probabilistic point of view. It will also be given the relationship between the heat kernel on ℂ and γ_{α, c_{1, 2}} (subordination property).

Keywords:

PACS: 46L53, 33D45, 44A15

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